18.S34 (FALL 2007) PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. (55P) Do there exist 1, 000, 000 consecutive integers each of which con tains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing n, we have that p2 divides n. Show that for any k � 1 there exist k consecutive integers, none of which is powerful. 3. Show that for any k � 1 there exist k consecutive positive integers, none of which is a sum of two squares. (You may use the fact that a positive integer n is a sum of two squares if and only if for every prime p � 3 (mod 4), the largest power of p dividing n is an even power of p.) 4. (56P) Prove that every positive integer has a multiple whose decimal representation involves all ten digits. 5. (66P) Prove that among any ten consecutive integers at least one is relatively prime to each of the others. 6. (70P) Find the length of the longest sequence of equal nonzero digits in which an integral square can terminate (in base 10), and ﬁnd the smallest square which terminates in such a sequence. 7. (72P) Show that if n is an integer greater than 1, then n does not divide 2n − 1. 8. Show that if n is an odd integer greater than 1, then n does not divide 2n + 2. � � �a� 9. (a) (77P) Prove that pa � b (mod p) for all integers p, a, and b pb with p a prime, p > 0, and a � b � 0. (b) (not on Putnam exam) Show in fact that the above congruence holds modulo p2 . (c) (not on Putnam exam) Show that if p � 5, then the above con gruence even holds modulo p3 . 1 10. (82P) Let n1 , n2 , . . . , ns be distinct integers such that (n1 + k)(n2 + k) · · · (ns + k) is an integral multiple of n1 n2 · · · ns for every integer k. For each of the following assertions, give a proof or a counterexample: (a) |ni | = 1 for some i. (b) If further all ni are positive, then {n1 , n2 , . . . , ns } = {1, 2, . . . , s}. 11. (83P) Let p be in the set {3, 5, 7, 11, . . .} of odd primes, and let F (n) = 1 + 2n + 3n2 + · · · + (p − 1)np−2 . Prove that if a and b are distinct integers in {0, 1, 2, . . . , p − 1} then F (a) and F (b) are not congruent modulo p, that is, F (a) − F (b) is not exactly divisible by p. 12. (85P) Deﬁne a sequence {ai } by a1 = 3 and ai+1 = 3ai for i � 1. Which integers between 00 and 99 inclusive occur as the last two digits in the decimal expansion of inﬁnitely many ai ? 13. (86P) What is the units (i.e., rightmost) digit of � 20000 � 10 ? 10100 + 3 Here [x] is the greatest integer � x. 14. (91P) Suppose p is an odd prime. Prove that � p � �� � p p+j � 2p + 1 (mod p2 ). j j j=0 15. (96P) If p is a prime number greater than 3 and k = ≥2p/3∈, prove that the sum � � � � � � p p p + +···+ 1 2 k of binomial coeﬃcients is divisible by p2 . 2 16. (97P) Prove that for n � 2, n terms n−1 terms ���� ···2 22 � ���� ···2 22 (mod n). 17. (99P) The sequence (an )n�1 is deﬁned by a1 = 1, a2 = 2, a3 = 24, and, for n � 4, 6a2n−1 an−3 − 8an−1 a2n−2 an = . an−2 an−3 Show that, for all n, an is an integer multiple of n. 18. (00P) Prove that the expression � � gcd(m, n) n n m is an integer for all pairs of integers n � m � 1. 19. (03P) Show that for each positive integer n, n! = n � lcm{1, 2, . . . , ≥n/i∈}. i=1 (Here lcm denotes the least common multiple, and ≥x∈ denotes the greatest integer � x.) 20. (04P) Deﬁne a sequence {un }� n=0 by u0 = u1 = u2 = 1, and thereafter by the condition that � � un un+1 det = n! un+2 un+3 for all n � 0. Show that un is an integer for all n. (By convention, 0! = 1.) 21. How many coeﬃcients of the polynomial � Pn (x1 , . . . , xn ) = (xi + xj ) 1�i<j�n are odd? 3 22. Deﬁne a0 = a1 = a2 = a3 = 1, an+4 an = an+3 an+1 + a2n+2 , n � 0. Is an is an integer for all n � 0? 23. Deﬁne a0 = a1 = 1 and n−1 an = 1 � 2 a , n > 1. n − 1 i=0 i Is an an integer for all n � 0? 24. Do there exist positive integers a and b with b − a > 1 such for every a < k < b, either gcd(a, k) > 1 or gcd(b, k) > 1? 25. Let f (x) = a0 + a1 x + · · · be a power series with integer coeﬃcients, with a0 ≤= 0. Suppose that the power series expansion of f ≤ (x)/f (x) at x = 0 also has integer coeﬃcients. Prove or disprove that a0 |an for all n � 0. 26. Suppose that f (x) and g(x) are polynomials (with f (x) not identically 0) taking integers to integers such that for all n ≡ Z, either f (n) = 0 or f (n)|g(n). Show that f (x)|g(x), i.e., there is a polynomial h(x) with rational coeﬃcients such that g(x) = f (x)h(x). 27. Let a and b be rational numbers such that an − bn is an integer for all positive integers n. Prove or disprove that a and b must themselves be integers. 28. Find the smallest integer n � 2 for which there exists an integer m with the following property: for each i ≡ {1, . . . , n}, there exists j ≡ {1, . . . , n} diﬀerent from i such that gcd(m + i, m + j) > 1. 4

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# 18.S34 (FALL 2007) DIVISIBILITY